Analysis of the paper by Kuleshov et al. (2018)
Project team: Piotr Pekala, Benjamin Yuen, Dmitry Vukolov, Alp Kutlualp
Problem statement: Proper quantification of uncertainty is crucial for applying statistical models to real-world situations. The Bayesian approach to modeling provides us with a principled way of obtaining such uncertainty estimates. Yet, due to various reasons, such estimates are often inaccurate. For example, a 95% posterior predictive interval does not contain the true outcome with 95% probability. Such a model is miscalibrated.
Context: why is this problem important or interesting? any examples?
Below we demonstrate that the problem of miscalibration exists and show why it exists for Bayesian neural networks in regression tasks. We focus on the following sources of miscalibration:
Our aim is to establish a causal link between each aspect of the model building process and a bad miscalibrated outcome.
Data Generation: We generate the data from a known true function with Gaussian noise. We then build multiple feedforward BNN models using:
Inference: We then obtain the posterior of the model by:
Diagnostics: We check for convergence using trace plots, the effective sample size, and Gelman-Rubin tests. In the case of variational inference, we track the ELBO during optimization. The simulated posterior predictive is evaluated visually.
The probabilistic library NumPyro provides fast implementations of both algorithms, which we make use of in this research. Due to time constraints we do not perform multiple random restarts, so the results may be subject to randomness.
Using a simple data-generating function $y_i = 0.1 x^3_i + \varepsilon$, where $\varepsilon \sim \mathcal{N}(0, 0.5^2)$ and a series of BNN models we evaluate the impact of our design choices on the posterior predictive.
A neural network with 50 nodes in a single hidden layer, well-chosen prior and noise, as well as correctly performed inference using sampling produce a posterior predictive that adequately reflects both epistemic and aleatoric uncertainty:
Naturally, our claims regarding the adequacy of epistemic uncertainty are subjective due to the absence of universal quantitative metrics.
The prior on the network weights defines epistemic uncertainty. A higher than necessary variance of the prior results in a significantly larger and most likely unreasonable epistemic uncertainty:
Lower variance of the prior prevents the model from adequately reflecting epistemic uncertainty in areas where no data is available. It also introduces bias: a neural network with 50 nodes in a single hidden layer (i.e. 151 weights) is unable to fit a cubic function:
The bias becomes apparent with an even narrower prior on the weights. This is a major issue with the model that needs to be fixed. Calibration is inappropriate in this case.
Similar to the previous example, a BNN may demonstrate bias by beeing too simple architecturally. That is difficult to demonstrate for a dataset generated by a cubic function (can be described by just 4 points). Still, we can try a combination of the prior and noise specification from the proper model and reduce the number of nodes. The sampler does not converge in this setup.
The appropriate level of the prior variance depends on the network complexity. A simpler network with 10 nodes and the same prior variance as our original benchmark model predicts much lower epistemic uncertainty. Therefore, the prior has to be selected for each particular network configuration.
The noise in the likelihood function corresponds to aleatoric uncertainty. The effect of the wrong noise specification is that aleatoric uncertainty is captured incorrectly. In the model below the noise is still Gaussian, but has a higher variance than the true one:
This might be a good candidate for calibration. Alternatively, one could find ways for the network to learn the noise from the data or put an additional prior on the variance of the noise.
Similarly, if the noise is too small, the resulting aleatoric uncertainty captured by the posterior predictive will be unrealistically low:
Using approximate methods of inference is also likely to lead to a miscalibrated posterior predictive. In the example below, Variational Inference with reparametrization on a network with 50 nodes produces too low epistemic uncertainty and slightly larger aleatoric uncertainty. Calibration may be used for correcting the latter.
To be filled...
Proposition: The authors of the paper propose a simple calibration algorithm for regression. In the case of Bayesian models, the procedure ensures that uncertainty estimates are well-calibrated, given enough data. In other words, the resulting posterior predictive aligns with the data, i.e. the observations fall within each posterior predictive interval (e.g. the 95% interval) with a corresponding probability.
Claim: The authors claim that the method outperforms other techniques by consistently producing well-calibrated forecasts, given enough i.i.d. data. Based on their experiments, the procedure also improves predictive performance in several tasks, such as time-series forecasting and reinforcement learning.
The algorithm has two main steps (from Algorithm 1 listing in the paper):
Construct a recalibration dataset $\mathcal{D}$: $$ \mathcal{D} = \left\{\left(\left[H\left(x_t\right)\right]\left(y_t\right), \hat P\left(\left[H\left(x_t\right)\right]\left(y_t\right)\right)\right)\right\}_{t=1}^T $$ where:
$$ \hat P(p)=\left|\left\{y_t\mid \left[H\left(x_t\right)\right]\left(y_t\right)\lt p, t=1\ldots T\right\}\right|/T $$
Suppose we have the following hypothetical posterior predictive, which is heteroscedastic and is underestimating uncertainty. For each value of the covariate $X$, the posterior predictive provides us with a conditional distribution $f(Y|X)$:
The first step of the calibration algorithm is to obtain predictive conditional distributions for each 𝑋 in the dataset. If no closed-form is available we simulate the posterior predictive based on the samples of the posterior:
An alternative, more commonly used notation for $H(x_t)$ is $F_t$ (a CDF)
The observed $Y$ for each data point falls somewhere within those conditional distributions. We evaluate the conditional CDFs at each observed value of the response $Y$ to obtain the predicted quantiles. In the absence of analytical form, we simply count the proportion of samples that are less than $y_t$. This gives us the estimated quantile of $y_t$ at $x_t$ in the posterior predictive distribution:
We next find the empirical quantiles, which are defined as the proportion of observations that have lower quantile values than that of the current observation. This is equivalent to finding the empirical CDF of the predicted quantiles. The mapping of predicted quantiles and the empirical quantiles will form a recalibration dataset:
The mapping is obtained for all observations in the dataset. Note that in this example the first two observations have different conditional distributions, but the same values of the predicted and empirical quantiles. The calibration procedure doesn't distinguish between such cases:
The inverse S-curve of the recalibration dataset in our example is characteristic of a posterior predictive that underestimates uncertainty. The diagnonal line denotes perfect calibration:
We then train a model (e.g. isotonic regression) on the recalibration dataset and use it to output the actual probability of any given quantile or interval. Here the 95% posterior predictive interval corresponds to a much narrower calibrated interval:
| Predicted quantiles | Calibrated quantiles |
|---|---|
| 0.025 | 0.224 |
| 0.5 | 0.477 |
| 0.975 | 0.776 |
Ideally, the model should be fit on a separate calibration set in order to reduce overfitting. Alternatively, multiple models can be trained in a way similar to cross-validation:
Concretely, for models without closed form posterior predictive CDF, the calibration algorithm is restated as:
As a visual diagnostic tool, the authors suggest using a calibration plot that shows the true frequency of points in each quantile or confidence interval compared to the predicted fraction of points in that interval. Well-calibrated models should be close to a diagonal line:
Several alternatives are available, each with specific advantages and disadvantages:
1. Calibration error $$cal(F_1, y_1, ..., F_N, y_N ) = \sum_{j=1}^m w_j \cdot (p_j − \hat{p}_j)^2$$ Provides a synthetic measure representing the overall 'distance' of the points on the calibration curve from the $45^\circ$ straight line. The weights ($w_j$) might be used to reduce the importance of intervals containing few observations. Value of $0$ indicates perfect calibration. Sensitive to binning.
2. Predictive RMSE $$\sqrt{\frac{1}{N}\sum_{n=1}^{N}||y_n-\mathbb{E}_{q(W)}[f(x_n,W)]||_{2}^{2}}$$ Measures the model fit to the observed data by normalizing the difference between observations and the mean of the posterior predictive. Minimizing RMSE does not guarantee calibration of the model.
3. Mean prediction interval width $$\frac{1}{N}\sum_{n=1}^{N}\hat{y}_{n}^{high} - \hat{y}_{n}^{low},$$ where $\hat{y}_{n}^{high}$ and $\hat{y}_{n}^{low}$ are - respectively - the 97.5 and 2.5 percentiles of the predicted outputs for $x_n$. Average difference between the the upper and lower bounds of predictive intervals evaluated for all the observations (different significance values might be used to define the predictive intervals). By itself provides information on the precision of the prediction (confidence with which a prediction is made) rather than calibration or miscalibration of the model. However may be used in conjunction with PICP.
4. Prediction interval coverage probability (PICP) $$\frac{1}{N}\sum_{n=1}^{N}\mathbb{1}_{y_n\leq\hat{y}_{n}^{high}} \cdot \mathbb{1}_{y_n\geq\hat{y}_{n}^{low}}$$ Calculates the share of observations covered by 95% (or any other, selected) predictive intervals. Alignment of the PICP with the probability mass assigned to the predictive interval generating it may misleadingly point to proper calibration if true noise distribution belongs to a different family than the posterior predictive. Requires a large sample of observations.
Rather than try to reproduce the experiments from the paper we chose to run the calibration algorithm on a series of synthetic datasets. This allows us to analyze the effect of the procedure on different purposefully miscalibrated posterior predictives.
The first dataset is a cubic polynomial with homoscedastic Gaussian noise:
Through sampling, we perform inference of a BNN that underestimates uncertainty (low variance of the noise in the likelihood). The calibration model is trained on a separated hold-out dataset of the same size. After calibration, the posterior predictive aligns with the data really well:
The calibrated posterior predictive isn't smooth due to sampling, which is especially evident for the extreme quantiles.
Similarly, excellent results are obtained when we apply the calibration algorithm to a BNN that overestimates uncertainty due to the high variance of the noise in the Gaussian likelihood function. The resulting posterior predictive captures aleatoric uncertainty well:
The next dataset is the one we used previously in our miscalibration examples — a third-degree polynomial with a gap in the middle. This will allow us to evaluate the impact of the calibration algorithm on epistemic uncertainty.
We sample from a BNN that produces a reasonably good posterior predictive, both in terms of aleatoric and epistemic uncertainty. After calibration, epistemic uncertainty shrinks, but only slightly. Since our definition of "good" epistemic uncertainty is subjective, the algorithm doesn't seem to ruin a valid model. However, epistemic uncertainty does become more asymmetric than before to calibration:
The same is true for the posterior predictives that either over- or underestimate uncertainty due to the wrong prior. The calibration algorithm has little effect on epistemic uncertainty:
The situation changes when the noise is specified incorrectly and there is missing data. Since the algorithm maps predicted quantiles to empirical ones uniformly across all input space, this calibration method produces perfect aleatoric uncertainty, but reduces epistemic uncertainty drastically:
Analogously, if aleatoric uncertainty is underestimated by the model, after recalibration, it will be aligned with the data as much as possible, while epistemic uncertainty will be blown up. The authors of the method explicitly state that the suggested approach only works given enough i.i.d. data. Here we see one instance of how it fails:
Another case of failure can be observed in the situation when there is bias, i.e. the network is not sufficiently expressive to describe the data due to a combination of the prior and the architecture. In an effort to fit the data the calibration algorithm increases uncertainty uniformly across the whole input space. A much better approach would be to change the bad model, rather than try to recalibrate it:
Correctly performed Variational Inference using isotropic Gaussians is often associated with underestimated epistemic uncertainty. The quantile-based calibration algorithm does little to such posterior predictives: both epistemic and aleatoric uncertainties mostly remain the same:
When the variance of the noise in the likelihood is specified incorrectly, the calibration method corrects that, aligning aleatoric uncertainty with the data. The resulting epistemic uncertainty, however, is again too low. The algorithm is unable to remedy the issues arising from variational approximation:
The next dataset is generated by the same third-degree polynomial, but with heteroscedastic noise that depends on the value of the predictor $X$. We will model it using a BNN with a constant variance of the noise in the likelihood, and see if the calibration procedure can fix the resulting miscalibrated posterior predictive.
When a BNN is unable to capture heteroscedastic noise, the quantile calibration only makes the posterior predictive worse. The central region of aleatoric uncertainty that the original posterior predictive captured correctly is now inflated. The resulting model might be producing better uncertainty on average, but is less precise in specific segments of the input space:
Our attempts to model heteroscedasticity by specifying constant noise in the likelihood function of a BNN so far were not very successful. We did observe oscillations of the prediction, but only on datasets with very few points or on artificially constructed datasets that do not occur in real-world scenarios:
The authors of the paper state that "if the true data distribution $P(Y | X)$ is not Gaussian, uncertainty estimates derived from the Bayesian model will not be calibrated". We will construct such a dataset by generating observations with Gamma noise, instead of Normal noise, fit an ordinary BNN to it and see how the proposed calibration algorithm performs:
The simulated posterior predictive obtained from a BNN with a Normal likelihood turns out to be indeed miscalibrated. All of the quantiles including the median are off. After applying the calibration procedure, all of the quantiles are aligned with the data. The non-parametric isotonic regression that lies at the core of the proposed calibration method seems to excel in this setting: